Question

Integrate the following function:
$\sin x\sin \left( {\cos x} \right).$

Answer 1

Hint: - Substitute the value of $\cos x = t$ and differentiate the equation with respect to x.

Let, $I = \int {\sin x\sin \left( {\cos x} \right)dx} $
Substitute, $\cos x = t.............\left( 1 \right)$
Differentiate equation 1 w.r.t. $x$
As we know differentiation of $\cos x = - \sin x$
$ \Rightarrow - \sin xdx = dt$
Substitute this value in the integral we have
$
  I = \int {\sin \left( t \right)\left( { - dt} \right)} \\
   \Rightarrow I = - \int {\sin tdt} \\
$
Now as we know integration of $\sin t$ is $- \cos t$
$ \Rightarrow I = - \left( { - \cos t} \right) + c$, where c is some arbitrary integration constant
Now put the value of $t$
$
   \Rightarrow I = \cos t + c \\
   \Rightarrow I = \cos \left( {\cos x} \right) + c \\
$
So, this is the required value of the integral.

Note: - In such types of question the key concept we have to remember is that always substitute some values to $t$ or any other variable, to make integration simple, then differentiate the variable you assumed w.r.t the given variable, then re-substitute this value in to integral, then always remember the basic differentiation and integration formulas, then simplify we will get the required value of the integral.